By the sound of it, you have multiple devices—wired and wireless—on your internal network. You need some kind of methodology to isolate the source of the problem first, otherwise you can waste a lot of time and energy taking random actions that will do nothing to halt it.
I suggest the following order of operations:
1. Positively identify all devices on the network. ie. log in to your router, look at the DHCP client table, and ensure that every client is indeed one of your household's devices. Cross-check the client name & IP address in the router's DHCP table against the actual hostname / IP address listed by the device. This can take some time, especially if you have many computers, laptops, tablets, phones, game consoles, etc.
2. Identify the data hog. If you have a router that provides realtime bandwidth monitoring and interface stats (ie. one running DD-WRT, Tomato, or something similar), then your job will be easy and you can probably isolate the problem just by looking at some of the stats on the various WAN and interface status screens.
If you're not so lucky, try downloading something like the free trial version of Capsa, and let it figure out what all the devices are, and roughly how much traffic they are generating:
http://www.colasoft.com/capsa/
Once you figure out what device is generating the most traffic, then you will have to sort out what process/service on that device is creating havoc. It would be difficult to give more specific advice without knowing what manner of device it is, but at the very least you can unplug that unit / disable its wireless access until you have a chance to examine it in depth and see what it's trying to do.
I think you need to read up on how the internet works.
If I was in between you and the email server, or the email server and another email server, or inbetween ..., then I could capture every packet, and if I can capture every packet I can read your email.
It is simple and easy. It isn't even 'hacker' stuff. It is general knowledge.
I'd rather recommend wireshark, but reading others emails is as easy as this. And don't think for a second encryption is secure either, unless you setup your keys on another network away from your primary network. I worked for a company for a while that does man in the middle to decrypt on [almost] all connections on business computers, by injecting a different key on both ends. It was just a single project in a business class router equivalent. MANY companies do this today and record their employees password and encrypted data without their consent and they never even suspect because, hey, it is https, right?
imho, don't check your banking, don't buy stuff from newegg or amazon, and generally avoid passwords like the plague at work. Trust me on this. If you have an evil sys admin or boss they will know your darkest secrets if you type it into your work computer, and it all starts with a password.
When I had a similar need, the free edition of Colasoft Capsa was the best I found. This is assuming you have a computer that can see all the traffic.